It can be shown that the second derivative of the first and the last term (\(J_i(WLS)\)  and \(J_i(CORR)\) ) leads to a positive definite matrix \(H = R^TQ_iR + \beta A_i\) , due to \(R^TQ_iR\)  being positive definite as all entries in \(Q_i\)  are greater than 0 (due to the infinite support of multivariate Gaussians defined in (REF )) and \(A_i\)  a symmetric matrix with entries in \([0,2]\)  — the addition of \(R^TQ_iR\)  and \(\beta A_i\)  leads to a matrix pencil, whose positive definiteness is granted whenever \(\beta > -1/\delta \)  with \(\delta \)  being the minimum positive eigenvalue of the generalized eigenvalue problem \(\lambda R^TQ_iR \mathbf {v} = A_i \mathbf {v}\)  [1]}; this is the case in our setting, because \(\beta \)  is always greater than 0 to embed (local) correlation-based learning.
Therefore, the function \(J_i(WLS) + J_i(CORR)\)  is convex, and due to the fact that i) the \(L_1\)  norm is convex and ii) a linear combination of two convex functions is again convex according to analysis made in [2]}, the objective function in (REF ) is convex.
